Q.1 Determine whether each of the following relations are reflexive, symmetric and
transitive:

(v) Relation R in the set A of human beings in a town at a particular time given by

(b) R = \{(x,y): x\;and\;y\;live\;in\;the\;same\;locality\}

Answers (1)

R = \{(x,y): x\;and\;y\;live\;in\;the\;same\;locality\}

\left ( x,x \right )\in R as x and x is same human being.So, it is reflexive.

\left ( x,y \right )\in R means  x\;and\;y\;live\;in\;the\;same\;locality.

It is same as y\;and\;x\;live\;in\;the\;same\;locality  i.e. \left ( y,x \right )\in R.

So,it is symmetric.

\left ( x,y \right ),\left ( y,z \right )\in R means  x\;and\;y\;live\;in\;the\;same\;locality and y\;and\;z\;live\;in\;the\;same\;locality.

It implies that x\;and\;z\;live\;in\;the\;same\;locality i.e. \left ( x,z \right )\in R.

Hence, it is reflexive, symmetric and
transitive.

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